Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C32 ⋊ #382 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C2×D32 |
| commutator | C22×C32 | C22×C4 |
| Fitting | C22×C32×C4 | C22 |
| socle, prime radical, cosocle | C23×C32 | C23 |
| hypercenter | C22×C4 | D32 |
| hypocenter | C32 | #382 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
362 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^2|a^2=d,b^2=d,c^2=ef,b^a=be,c^a=cf,h^a=h^2,c^b=cf,g^b=g^284 conjugacy classes — view the full character table.
| maximal subgroup of | #11770925, #11771072, #11775363, … |
| maximal normal subgroup of | #11770925, #11771072, #11775363, … |
| maximal quotient of | #11773189, #11773191, #11773211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 3242);